Inverse Trigonometric Functions Question 58
Question: The value of the expression $ {{\sin }^{-1}}( \sin \frac{22\pi }{7} ) $
$ {{\cos }^{-1}}( \cos \frac{5\pi }{3} ) $ + $ {{\tan }^{-1}}( \tan \frac{5\pi }{3} ) $ + $ {{\sin }^{-1}}(cos2) $ is
Options:
A) $ \frac{17\pi }{42}-2 $
B) $ -2 $
C) $ \frac{-\pi }{21}-2 $
D) none of these
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Answer:
Correct Answer: A
$ {{\sin }^{-1}}\sin ( \frac{22\pi }{7} )={{\sin }^{-\text{1}}}\sin ( 3\pi +\frac{\pi }{7} )=\frac{\pi }{7} $
$ {{\cos }^{-1}}\cos ( \frac{5\pi }{3} )={{\cos }^{-1}}\cos ( 2\pi -\frac{\pi }{3} )=\frac{\pi }{3} $
$ {{\tan }^{-1}}\tan ( \frac{5\pi }{7} )={{\tan }^{-\text{1}}}\tan ( \pi -\frac{2\pi }{7} )=\frac{5\pi }{7} $
$ {{\sin }^{-1}}\cos (2)=\frac{\pi }{2}-{{\cos }^{-1}}\cos 2=\frac{\pi }{2}-2 $
$ \therefore $ Required value $ =-\frac{\pi }{7}+\frac{\pi }{3}-\frac{2\pi }{7}+\frac{\pi }{2}-2 $
$ =\frac{(-18+35)\pi }{42}-2 $
$ =\frac{17\pi }{42}-2 $
BETA
